Question

Show that if P and Q are vector Subspaces of a vector space V then P ∩ Q is also a vector subspace of V.

Answer 1

Let $x,y\in P\cap Q$ and $\alpha \in \mathbb{F}$. We shall show that $P\cap Q\neq \phi$ $\alpha(x+y) \in P\cap Q$

$\text{Since P and Q are subspaces of V then, P }≠\phi \text{ and } Q≠\phi\implies P\cap Q\neq \phi$

$\text{Since P and Q are subspaces of V }\implies \alpha(x+y) \in P \text{ and } \alpha(x+y) \in Q$

Thus, $\alpha(x+y) \in P\cap Q$

Hence, $P\cap Q$ is a subspace of $V$